Question

Cos5x=1/2 Решение уравнения

Answer 1

Ответ:

$x= \pm \frac{\pi}{15}+\frac{2\pi}{5}n,n\in\mathbb{Z}$

Пошаговое объяснение:

$\cos 5x=\frac{1}{2}\\5x=\pm \frac{\pi}{3}+2\pi n,n\in\mathbb{Z}\\x= \pm \frac{\pi}{15}+\frac{2\pi}{5}n,n\in\mathbb{Z}$

Answer 2

To solve the equation cos(5x) = 1/2, we need to find the values of x that satisfy this equation. Remember that the cosine function has periodicity, so there are multiple solutions within a given range. Let's find the solutions for 0 ≤ x ≤ 2π (one full period of the cosine function):

1. Find the angles where cos(θ) = 1/2: We know that cos(π/3) = 1/2. Therefore, one solution is: θ₁ = π/3

   However, the cosine function also has another angle where it is equal to 1/2 in the first quadrant: θ₂ = 2π - π/3 = 2π/3

2. Since the cosine function has a period of 2π, we can find additional solutions by adding integer multiples of the period to the previous solutions:

   θ₃ = θ₁ + 2π = π/3 + 2π = 7π/3 θ₄ = θ₂ + 2π = 2π/3 + 2π = 8π/3

3. Now, we have four solutions for x: π/3, 2π/3, 7π/3, and 8π/3.

However, the original equation cos(5x) = 1/2 has an extra constraint due to the coefficient 5 in front of x. For any integer n, we can use the following equation:

5x = θ + 2nπ, where θ is one of the solutions found earlier.

To get the values of x, divide both sides of the equation by 5:

x = (θ + 2nπ) / 5

Now, plug in each of the values of θ and n to get all the solutions:

1. When θ = π/3 and n = 0: x₁ = (π/3) / 5

2. When θ = 2π/3 and n = 0: x₂ = (2π/3) / 5

3. When θ = 7π/3 and n = 0: x₃ = (7π/3) / 5

4. When θ = 8π/3 and n = 0: x₄ = (8π/3) / 5

These are the solutions to the equation cos(5x) = 1/2 for 0 ≤ x ≤ 2π. Note that you can find infinitely many solutions by varying the value of n (integer).